Problem: $f(x,y) = y^2\tan(x)$ What is $\dfrac{\partial f}{\partial x}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $0$ (Choice B) B $2y\sec^2(x)$ (Choice C) C $2y\tan(x)$ (Choice D) D $y^2\sec^2(x)$
Answer: We want to find $\dfrac{\partial f}{\partial x}$, which is the partial derivative of $f$ with respect to $x$. When we take a partial derivative with respect to $x$, we treat $y$ as if it were a constant. Here, $f(x, y)$ only has one term. $\begin{aligned} &\dfrac{\partial}{\partial x} \left[ y^2\tan(x) \right] = y^2 \dfrac{\partial}{\partial x} \left[ \tan(x) \right] = y^2\sec^2(x) \end{aligned}$ In conclusion: $\dfrac{\partial f}{\partial x} = y^2\sec^2(x)$